Radha and Hema are neighbours who cycle to the same school along the same route. Radha's cycling speed is 8 km/h, and Hema's cycling speed is 10 km/h. If Hema takes exactly 9 minutes less than Radha to reach the school, how far is the school from their locality?

Introduction / Context:
This is a straightforward time and distance problem comparing two people traveling the same route to school at different speeds. The given difference in travel times allows us to calculate the actual distance. This kind of question is very common in aptitude exams to test basic understanding of the relation time = distance / speed.

Given Data / Assumptions:

• Radha's speed = 8 km/h.
• Hema's speed = 10 km/h.
• Both travel from the same locality to the same school.
• Hema takes 9 minutes less than Radha.
• 9 minutes = 9 / 60 hours = 0.15 hours.
• We assume constant speeds for both girls.

Concept / Approach:
Let the distance to school be D km. The time taken by each is distance divided by speed. The question gives the difference in these times. We set up an equation using the difference in times and solve for D. Then, we check which option matches this value. This process uses the fundamental formula time = distance / speed in an algebraic way.

Step-by-Step Solution:
Let D be the distance to the school in km. Time taken by Radha = D / 8 hours. Time taken by Hema = D / 10 hours. Hema takes 0.15 hours less, so D / 8 - D / 10 = 0.15. Compute the left side: D * (1 / 8 - 1 / 10) = D * (10 - 8) / 80 = D * (2 / 80) = D / 40. So D / 40 = 0.15. Therefore, D = 0.15 * 40 = 6 km.

Verification / Alternative check:
Check the times using D = 6 km. Radha's time = 6 / 8 = 0.75 hours = 45 minutes. Hema's time = 6 / 10 = 0.6 hours = 36 minutes. The difference is 45 - 36 = 9 minutes, which exactly matches the given condition. This confirms that the distance is correctly calculated.

Why Other Options Are Wrong:
Distances of 5 km, 5.5 km, 6.5 km, and 4.5 km do not produce a time difference of exactly 9 minutes when substituted into D / 8 and D / 10. They yield different time differences and therefore cannot satisfy the condition stated in the problem. Only 6 km produces the correct 9 minute difference in travel time.

Common Pitfalls:
Students sometimes mistakenly subtract speeds instead of computing the difference in times. Others may forget to convert minutes into hours properly before forming the equation. Carefully converting 9 minutes to 0.15 hours and applying the formula time = distance / speed helps avoid these errors and leads to the correct result.

Final Answer:
The distance from their locality to the school is 6 km.